Completeness of the set of scattering amplitudes

Abstract

Let $f\in L^2(S^2)$ be an arbitrary fixed function with small norm on the unit sphere $S^2$, and $D\subset {\mathbb{R}}^3$ be an arbitrary fixed bounded domain. Let $k>0$ and $\alpha \in S^2$ be fixed. It is proved that there exists a potential $q\in L^2(D)$ such that the corresponding scattering amplitude $A({\alpha }^{\prime })=A_q({\alpha }^{\prime })=A_q({\alpha }^{\prime },\alpha ,k)$ approximates $f({\alpha }^{\prime })$ with arbitrary high accuracy: $\|f(α')-A_q(α')_{L^2(S^2)}\|\leq\ve$ where $\ve>0$ is an arbitrarily small fixed number. This means that the set $\{A_q({\alpha }^{\prime }){\}}_{\forall q\in L^2(D)}$ is complete in $L^2(S^2)$. The results can be used for constructing nanotechnologically "smart materials".

Cited by in corpus (1)

• Electromagnetic wave scattering by many small particles